Vapour pressure Kelvin equation

In this model also the decrease of the pore radius due to the formation of an adsorbed layer is incorporated. Flow 1 in Fig. 9.9 is the case of combined Knudsen molecular diffusion in the gas phase and multilayer (surface) flow in the adsorbed phase. In case 2, capillary condensation takes place at the upstream end of the pore (high pressure Pi) but not at the downstream end (P2), and in case 3 the entire capillary is filled with condensate. The crucial point in cases 3 and 4 is that the liquid meniscus with a curved surface not only reduces the vapour pressure (Kelvin equation) but also causes a hydrostatic pressure difference across the meniscus and so causes a capillary suction pressure Pc equal to... 

The equilibrium vapour pressure, P, over a curved surface is defined by tlie Kelvin equation... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

From the Kelvin equation it follows that the vapour pressure p over a concave meniscus must be less than the saturation vapour pressure p°. Consequently capillary condensation of a vapour to a liquid should occur within a pore at some pressure p determined by the value of r for the pore, and less than the saturation vapour pressure—always provided that the meniscus is concave (i.e. angle of contact <90°). 

The formation of a liquid phase from the vapour at any pressure below saturation cannot occur in the absence of a solid surface which serves to nucleate the process. Within a pore, the adsorbed film acts as a nucleus upon which condensation can take place when the relative pressure reaches the figure given by the Kelvin equation. In the converse process of evaporation, the problem of nucleation does not arise the liquid phase is already present and evaporation can occur spontaneously from the meniscus as soon as the pressure is low enough. It is because the processes of condensation and evaporation do not necessarily take place as exact reverses of each other that hysteresis can arise. 

Equation (6.50) is often referred to as the Thomson s (or Kelvin s) equation. As an example of the effect of this equation, the vapour pressure of a spherical droplet of molten Zn at the melting temperature is shown as a function of the droplet radius in Figure 6.14. 

This expression, known as the Kelvin equation, has been verified experimentally. It can also be applied to a concave capillary meniscus in this case the curvature is negative and a vapour pressure lowering is predicted (see page 125). 

The Young-Laplace equation (3.4/3.5) shows that, pA>pB, the pressure inside a bubble or drop exceeds that outside. For a sphere, Ap=pA - Pb = 2y/R, so that Ap varies with the radius, R. Thus the vapour pressure of a drop should be higher, the smaller the drop. This is shown by a related equation, the Kelvin equation [13,26], which is described here. 

Most models to calculate the pore size distributions of mesoporous solids, are based on the Kelvin equation, based on Thomson s23 (later Lord Kelvin) thermodynamical statement that the equilibrium vapour pressure (p), over a concave meniscus of liquid, must be less than the saturation vapour pressure (p0) at the same temperature . This implies that a vapour will be able to condense to a liquid in the pore of a solid, even when the relative pressure is less than unity. This process is commonly called the capillary condensation. 

A simple derivation of the Kelvin equation is presented by Broekhoff and van Dongen [16]. Imagine a gas B in physical adsorption equilibrium above a flat, a convex, and a concave surface, respectively (see Fig. 12.8). Considering a transfer of dN moles of vapour to the adsorbed phase at constant pressure and temperature, equilibrium requires that there will be no change in the free enthalpy of the system. 

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

Kelvin s equation for surface energy and temperature, 138 for vapour pressure over a curved surface, 366 Khaikin s viscosity equation, 107 Kistiakowsky s equation for specific cohesion, 152, 162... 

This technique, developed by Eyraud [140] modified by Katz et al. [143] and recently by Cuperus et al. [141], is based on the controlled blocking of pores by capillary condensation of a vapour (e.g. CCli, methanol, ethanol, cyclohexane), present as a component of a gas mixture, and the simultaneous measurement of the gas flux through the remaining open pores of the membrane. The capillary condensation process is related to the relative vapour pressure by the Kelvin equation. Thus for a cylindrical pore model and during desorption we have... 

The drying rate decreases for several reasons. In the first place the vapour pressure Py above a concave liquid meniscus with radius Pp decreases during the process. According to the Gibbs-Thomson (or Kelvin) equation Py is given by Eq. (8.4) ... 

This has been explained as being due to capillary condensation in the pores of the adsorbent. As the pressure is reduced, the adsorbate does not evaporate as readily from the capillaries as it does from a flat surface due to a lowering of the vapour pressure over the concave meniscus formed by the condensed vapour in the pores. The lowering of the vapour pressure (p) for a cylindrical capillary of radius rj is given by the Kelvin equation ... 

Equation (1) is a useful route to calculating headspace concentrations above a pure substance from vapour pressures c is the gas phase concentration in g 1 1, p° is the saturated vapour pressure in mmHg, and T is the temperature in Kelvin. Equation (2) is the same equation restated in terms of concentration m in mol 1 1 at a temperature of 25 °C for cases where the molecular mass is unknown. These equations derive directly from the ideal gas equation. 

The vapour pressure of small crystals was considered from the point of view of the Kelvin equation by Pawlow,5 and Frumkin and Fuchs.6 Balarew and Kolarov7 supposed that it reached a minimum value with decreasing size.. ... 

The vapour pressure over a curved surface, p, is related to the vapour pressure over a flat surface, po, by the Kelvin equation ... 

Sintering does not usually involve chemical reactions, and the driving force is a reduction in the surface area and the associated reduction in surface energy. This driving force can be illustrated for a flat surface that contains a spherical promberance and a similar spherical depression. The vapour pressure over a curved surface is related to the vapour pressure over a flat surface by the Kelvin equation. This shows that the vapour pressure over a protuberance will be greater than the vapour pressure over the flat surface and will increase as the radius of the curved surface decreases. Similarly, the vapour pressure over a depression will be less than the vapour pressure over a flat surface. When a solid is heated, vapour transfer of matter will take place from a protuberance to a depression, and the surface will tend to become flat. 

Liquid bridges will coexist with adsorbed Aims and thus equilibrium may only be established if at the edge of the bridge the partial pressure p given by the Kelvin equation is equal to the resultant partial pressure p, at the Arst adsorbed layer on the solid surface. To obtain equilibrium, mass transport of vapour between the bridge surface and the monolayer would occur and the bridge would either grow or evaporate until equilibrium was reached. 

Various practical isotherm equations have been presented and they are useful in describing adsorption data of many adsorption systems. Among the many equations presented, the Toth equation is the attractive equation because of its correct behaviour at low and high loading. If the Henry behaviour is not critical then the Sips equation is also popular. For sub-critical vapours, multilayer isotherm equations are also presented in this chapter. Despite the many equations proposed in the literature, the BET equation still remains the most popular equation for the determination of surface area. When condensation occurs in the reduced pressure range of aroimd 0.4 to 0.995, the theory of condensation put forward by Kelvin is useful in the determination of the pore size as well as the pore size distribution. 

I hc vapour pressure Pr over a liquid drop of a radius r (the system of disconnected segments ) is known to exceed that over a flat surface of the same liquid, which is reflectrxl in Thomson s (Kelvin s) equation... 

Calculated from the equation ln(H) = 11.04 —5196/T + 0.03998 1, where I is the ionic strength in molarity and T is the temperature in degree kelvin. This equation was obtained by linear regression of the data from Imakawa, H., Chemical reactions in the chlorate manufacturing electrolytic cell (part 1) the vapour pressure of hypochlorous add on its aqueous solution, J. Electrochem. Soc. Jpn., 18, 382, 1950 and Imagawa, H., Studies on chemical reactions of the chlorate cell (part 2) the vapour pressure of hypochlorous acid on its mixed aqueous solution with sodium chlorate, J. Electrochem. Soc. Jpn., 19, 271,1951. 

Equation (3.6) is called Young-Laplace equation, in which R is the harmonic mean of the principal radii of curvature. The capillary pressure promotes the release of atoms or molecules from the particle surface. This leads to a decrease of the equilibrium vapour pressure with increasing droplet size Kelvin equation) ... 

Consider a spherical droplet of pure solid or liquid in contact with its vapour. The equilibrium is given by Equation (9.2), but it is now necessary to take account of the surface contribution to the total free energy. The equilibrium vapour pressure, pp, under these circumstances depends on the radius of curvature, r, of the surface according to the Kelvin equation ... 

The vapour pressure of the capillary water above the meniscus which is formed in the pores, is a function of the pore radius. It is not necessarily the same as the geometrical radius of the pores at the point where the meniscus is. A layer of water adheres to the pore waUs during dehydration of the pores [334]. C. Piebce [532] has made a compilation of experience in 4 his connection, with the help of which the radius calculated from the Kelvin equation can be converted into the efifective pore radius. The sizes of larger pores can be determined with the electron microscope. The average silica gel preparations have a mean pore radius of the order of 25 A. 

The capillary condensation phenomenon was discovered by Zsigismody [139], who investigated the uptake of water vapour by silica materials. Zsigismody proved that the condensation of physicosorbed vapours can occur in narrow pores below the standard saturated vapour pressure. The main condition for the capillary condensation existence is the presence of liquid meniscus in the adsorbent capillaries. As it is known, the decrease of saturated vapour pressure takes place over the concave meniscus. For cylindrical pores, with the pore width in the range 2-50 nm, i.e., for the mesopores, this phenomenon is relatively well described by the Kelvin equation [14]. This equation is still widely applied for the pore size analysis, but its main limitations remain unresolved. Capillary condensation is always preceded by mono- and/or multilayer adsorption on the pore walls. It means that this phenomenon plays an important, but secondary role in comparison with the physical adsorption of gases by porous solids. Consequently, the true pore width can be assessed if the adsorbed layer thickness is known. 

Kelvin Equation. An equation useful in sorption studies for the calculation of pore size and pore size distribution. It is rRTln(p/po) = -2yV cos 0 where p is the equilibrium vapour pressure of a curved surface (as in a capillary or pore) of radius r p is the equilibrium pressure of the same liquid on a plane surface R is the gas constant T is the absolute temperature y is the surface tension V the molar volume 0 the contact angle of the adsorbate. When the Kelvin equation is satisfied, vapour will condense into pores of radius r. (W.T. Thompson, Phil. Mag. 42,1871, p.448). 

The surface of a liquid assumes different shapes in different situations. When water is in a pipette, it has a concave surface. When on a glass slide as a fQm, it has a plane surface, and in the form of drops it has a convex surface (see Figure 6.4). When liquid from a plane surface forms a drop, there is an increase in surface area. This increase requires more surface energy. The molecules of the convex surface are attracted by other liquid molecules to a less extent now than they were in a plane surface. The drops also have a higher vapour pressure than earlier (plane surface). The opposite is observed for a concave liquid surface. There, the surface liquid molecules are attracted more than they are in a plane surface. The vapour pressure here is also the lowest amongst the three surfaces. These observations are compiled in an equation given by Kelvin. 

The Kelvin equation indicates that liquid drops have a considerably higher vapour pressure than bulk liquid. 

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